A studying on load transfer in carbon nanotube/epoxy composites under tension

1 Introduction

Applications of carbon nanotubes (CNTs) as nanoscale reinforcements in composites have been widely studied owing to their unusual properties [1]. Conventional reinforcing fibers, such as carbon fibers, are brittle and prone to break at elongations. CNTs, instead, are exceptionally resilient, and their elastic range is remarkably large. It has been known that mechanical properties such as stiffness and durability of composites could be enhanced by introductions of CNTs to the matrix. It has been revealed that load transfer from matrix to CNTs plays an important role in enhancing mechanical properties of composites. Numerous studies on load transfer have been carried out for identifying mechanical properties of composites [2–8]. Schadler et al [2]. experimentally investigated the load transfer from epoxy matrix to nanotubes using the Raman spectroscopy. Wherein, the multi-walled CNT (MWCNT)/epoxy composites are subjected to both tension and compression. Authors reported that the load transfer in the composites under tension is much higher in compression. Gao et al [3]. developed a shear-lag model for CNT-reinforced polymer composites to study the load transfer from polymer matrix to inner CNTs layers and factors affecting the load transfer using a multiscale approach. The numerical results reveal that the aspect ratio of the length to the diameter of CNTs is critical in load transfer in nanotube-reinforced composites. The predictions by the new model were compared favorably with the existing computational and experimental results. An analytical study was also conducted to investigate load transfer in single-walled CNT (SWNT)-reinforced polymer matrix composites [4], in which a simplified 2D representative volume element (RVE) of the CNT composites was considered. An expression of the effective length of CNTs was established for studying load transfer in CNT-reinforced composites. The effects of the CNT aspect ratio, CNT volume fraction, and matrix modulus on the axial stress and interfacial shear stress were also analyzed with a validation study from finite element method (FEM). Recently, a new shear-lag model was proposed to investigate load transfer in MWCNTs [5]. Results showed that SWCNTs exhibit the SWCNTs exhibit a greater load transfer efficiency than MWCNTs associated with the same CNTs volume fraction in the nanocomposites. Moreover, the incompetent behaviors of MWCNTs would become more substantial with an increase in the CNT layers. A micromechanical hybrid finite element approach to study the stress transfer in SWCNT-reinforced composites was developed [6]. The effects of CNT volume fraction, interfacial stiffness, and elastic modulus of matrix on load transfer from matrix to SWCNTs were analyzed. A method for evaluating elastic properties of the interfacial region was developed by examining fracture behaviors of CNT-reinforced poly (methyl methacrylate) (PMMA) matrix composites under tension by using molecular dynamics simulations [7]. The effects of the aspect ratio of CNT reinforcements on elastic properties such as Young’s modulus and yield strength of the interfacial region and the nanotube/polymer composites were investigated. Using the shear-lag model to investigate the influence of the volume fraction and matrix modulus on stress transfer was carried out [8]. There are few studies from available studies using shear-lag model and finite element method on load transfer in CNT-reinforced composites associated with more than 5 CNT layers due to the complicate nature of the problem involving many CNT layers. As a result, a development of a new theoretical model for load transfer in the composites associated with numerous CNT layer numbers is critical in understanding the potential of CNT composites. Answers to some critical issues by the new model such as the effect of CNT layer numbers and aspect ratio [5, 9] are indispensable in applications of CNT-reinforced composites. Furthermore, it is expected that the theoretical model can provide a comprehensive investigation on effects of matrix properties on load transfer since the thickness and Young’s modulus of matrix and shear modulus magnitude of interface between matrix and reinforcement would provide a more effective tool in understanding load transfer in the composites.

The load transfer between two adjacent CNT layers is strongly dependent on the interface interaction as well as interface properties between them [10]. Thus, the load transfer between epoxy matrix to inner CNT layers exists due to the presence of interfacial interaction between outermost CNT layer with matrix. In this work, a new shear-lag model for 2-D REV is originated based on a model of published literature [11] to investigate effects of epoxy and CNT properties and sizes on load transfer in CNT/epoxy composites subjected to a tensile load. In addition to the effect of the CNT layer number and aspect ratio of MWCNTs (AR = L/R_i.1) on load transfer, studies on effects of thickness and Young’s modulus of epoxy and interface bonding stiffness between epoxy and CNT layers are fully covered. The shear-lag model is validated by FEM and the Haque’s model [4]. Composites associated from 1 to 10 CNT layers are examined.

2 Experimental background

Figure 1 shows the importance of load transfer in CNTs in understanding crack growth in composites under tension [12]. It is observed from Figure 1 that by folding this thin composite, an opening crack was incurred at the maximum bending site. The SEM micrographs reveal the responses of the tensioned CNTs. Most CNTs were pulled out from the crack surface, while some were in position to bridge the crack, both being the important mechanisms of hindering crack growth for this material. The tensile load was transferred from the epoxy to the CNT outer-layer and then into the inner layers. The effectiveness of load transfer into CNTs affects the strengthening and toughening by CNTs for this nano-composite. There are three kinds of CNT failure modes in its composites, i. e., completely fracture, outer-most wall fracture, and pull out refer to [13]. This work focuses on the pull-out mode. Feasibility of this work is pullout mode and there are two conditions satisfied as (a) the maximum stress on each walls is less than the strength of CNTs (otherwise wall fracture happens); (b) the inter-wall shear stress is less than the inter-wall shear strength (otherwise wall sliding happens).

Figure 1:

SEM micrographs of the fracture surface of a CNT/epoxy composite. (a) Side view of a crack tip showing CNT bridging and pullouts. (b) A front view showing pullout holes left in the epoxy (arrows). Both reveal the importance of load transfer in CNTs to resist crack growth in composites.

3 Shear-lag model

Figure 2 illustrates the shear-lag model of a three-walled CNT/epoxy composite under tensile load. In order to study load transfer inside the CNT/epoxy composite, the atomic discrete structure of CNTs is converted into a continuum structure, in which the shear-lag model is adopted [10, 14, 15]. In the shear-lag model, the variation of stress on the thin CNT layers in the transversal direction is negligible and only the stress in the longitudinal direction is considered [5, 10]. The parameters used in this present work are shown in Figure 2, wherein L, R_i.0 and R_i.1 are half of the CNT length, RVE length, and the outer and inner radii of the epoxy, respectively. Index t = R_i.0−R_i.1 indicates the thickness of the epoxy in x-direction. The 2-D RVE structure is constructed symmetrically with respect to both x and y coordinate axes. In the study, the perfect bonding between CNT and epoxy is assumed and the thickness of this interface can be negligible [5, 8]. Radii, R_i.1 ~R_i.2n are outermost and innermost radii of CNTs associated with n layers, respectively. In the study, individual CNT thickness, p, is selected as 0.14 nm [5, 10, 14, 15] and the gap between two adjacent layers is s = 0.34 nm [9]. The inner radius of the innermost layer, n, is firmly chosen as R_i.2n= 5 nm. Subsequently, the outer and inner radii of CNT layer are R_i.2n-1= R_i.2n+ 0.14 nm and outer and inner radii of the gap between two adjacent layers are defined as R_i.2f-2= R_i.2f-1+ 0.2 nm, where f is the interface number index. The abovementioned radius relationships are applied to the REV with n CNT layers. An initial stress, σ_0m, is applied to the matrix [5, 8]. For simplicity, the 3-D model of the cylindrical REV is simplified into a 2-D model with all properties of the 3-D model remaining the same in the 2-D model. In the model, the gap between two adjacent CNT layers is also treated as an entity governed by the van der Waals interaction modeled by an interface modulus [5, 10]. The interface modulus between two adjacent layers denoted as G_iscb is still under debate and an accurate value of interface modulus between two adjacent CNT layers is under research. The available interface shear modulus was proposed as G_iscb= 4.2 GPa by neutron spectrometry [16]. Based on the shear-lag model, one infinitesimal element of the layer 2 with a size of dy is extracted to examine the force equilibrium. The axial stresses of the epoxy and CNT layers 1, 2, and 3, are denoted as σ_m, σ₁, σ₂, and σ₃, respectively. The three-layer model in Figure 2 can be extended to a model containing a larger layer number. It is noted that the axial stress of the cross-section of the epoxy is a function of x. In order to facilitate the shear-lag model, the average of the axial stress in the epoxy is defined to be σ_a.m [3, 8]:

Figure 2:

Shear-lag model for a three-walled CNT/epoxy composite under tension.

The equilibrium on the epoxy layer can be written as

The stress of the outermost CNT layer 1 is then expressed by

where A₁ is the cross sectional area of the CNT layer 1. For other internal layers, the force equilibrium can be obtained as

where k indicates the layer index, and σ_k, R_i.k and A_k are the axial stress, inner radius, and cross sectional area of CNT layer k, respectively. Deformation of the CNT layer k is governed by two interfacial shearing tractions, τ_i.k and τ_i.k + 1 on its inner and outer surfaces.

For the innermost layer (k = n), there is only one interface shearing traction τ_i.n Thereby the force equilibrium equation for this layer is provided by:

where n is the total shell numbers of the MWCNT .

In the study, the interfacial shear stress governing the facial sliding between two adjacent layers is assumed to be proportional to the difference of two axial strains of the two adjacent layers. Hence, the following relation for the interfacial shear stress is obtained:

τ_i.j is the interface shear stress of the interface j and ε_j and ε_j + 1 are the axial strains of the layer j and layer j + 1, respectively. R_i.j is the inner radius of the layer j and R_a.(1,n) is an average radius of the layer 1 and n. G_is* represents either the interface shear modulus between the matrix and CNT layer, G_ismat, or the modulus between two adjacent CNT layers, G_iscb.

By substituting eq. 6 into eqs (2)–(5) and expressing them in term of axial strains, we have:

where, E_mat and E_cb are Young’s module of the epoxy and CNTs, respectively.

Finally, the global force equilibrium of the CNT/epoxy composites is described in terms of the strain of the n-layered CNT/epoxy composite through a system of n + 1 differential equations. The boundary conditions at the edge y = L are provided as:

The system of the n first order differential eqs (7)–(10) describing the stress transfer from the epoxy to interior CNT layers is hence obtained. In order to calculate the stresses, the Fourth-order Runge-Kutta algorithm, RK-4 [17], is utilized. The integration is carried out from the edge toward the center of the system. After obtaining the strains, the axial stresses σ_k can be derived by magnifying the obtained strains by the relationship, σ_m = E_*.ε_m, where E_* may indicate either E_mat or E_cb. In the present work, numerical solutions are obtained for cases of n = 1–10. CNTs with other layer numbers can be obtained from adding and removing the outermost CNT layers successively. Young’s modulus of CNT is selected as E_cb= 1 TPa. The atomistic interaction effect between adjacent CNT layers results in the interface shear modulus G_is. In this work, the value of G_is= 4.2 GPa [16] and E_mat= 20 GPa of glass filled epoxy resin are used, unless otherwise noted. The initial stress applied to the epoxy is selected as σ_0m= 1 GPa. Epoxy thickness is set to be t = 5 nm. The shear modulus of the interface between the epoxy and CNTs is selected as G_ismat= G_mat= 10 GPa which is equivalent to Young’s modulus of epoxy resin, E_mat= 20 GPa.

4 Finite element method

To validate the shear-lag model, the finite element software ANSYS workbench, is used. In ANSYS simulations, each CNT layer is treated as a continuum thin layer [5, 10]. The original 3-D structure of the cylindrical REV is used in simulations as shown in Figure 3. To simulate the extent of interface interaction between two adjacent CNT layers, a thin layer representing the effective interphase is introduced between two edges of two adjacent CNT layers with a thickness analogues to that of the theoretical analysis, i. e. 0.2 nm. All parameters such as radii, length, boundary condition, material properties and interfacial bonding used in ANSYS simulations are selected the same with those in the shear-lag model. Poisson’s ratio, v is selected as 0.006 [18]. In addition, one more element is implemented in ANSYS as the interphase, which represents the interface interaction between two adjacent CNT layers. The interphase layer is also assumed to be isotropic. Hence, the interphase Young’s modulus in individual CNT interface is defined as E_iscb= 2.G_iscb.(1 + v) in the ANSYS analysis.

Figure 3:

A 3-D model of REV of the CNT/epoxy composite in finite element software, ANSYS.

5 Results and discussions

FEM is firstly used to validate the findings based on the shear-lag model. Figure 4 shows the axial stress distributions of the epoxy and the layer 1 of a five-walled CNT/epoxy composites at the aspect ratio AR = 10 and distance s = 0.34 nm subjected to a tensile loading obtained from the shear-lag model and FEM. The results show that induced stress distributions in the layer and epoxy from both predictions are in a very reasonable agreement in general. However, still small differences can be found on saturated stress lengths of the CNT layer and the epoxy. A saturated stress length is a key component representing the load transfer in the CNT composites that is denoted as L_ss and defined as a region covered by the saturated stress which is set as σ_s. From Figure 4, the saturated stress length ends up at y/L = 0.5 and 0.6 found by the FEM and shear-lag model, respectively which indicates that the saturated stress length by the FEM is 5 nm longer than that by the shear-lag model. In addition, the difference of the normalized saturated stress value, σ_s /σ_0m from the two models is observed around 4 %. In calculations by FEM, an element is built to represent the shear rigidity between CNT layers. Hence, a small amount of load is distributed in the element. On the contrary, the shear-lag model does not take into account the load bearing effect by the interface layer, but only considers the interface shear traction and the gap between two CNT adjacent layers. Therefore, the small difference of the results by the two approaches is mainly caused by modeling procedures.

Figure 4:

Axial stress distribution in a five-walled CNT and epoxy obtained from the shear-lag model and FEM subjected to a tension, AR = 10.

Results on the stress distribution in the CNT layer and load transfer from the shear-lag model in an SWCNT/epoxy composite at AR = 100 and distance s = 0.34 nm are further compared with that from the Haque’s analytical model as shown in Figure 5. Results from Haque’s work [4] at AR = 100 are used for the comparison. Based on Figure 5, the saturated stress length ends up at y/L = 0.925 and 0.98 by the Haque’s model and shear-lag model, respectively. Hence, it is concluded that stress saturation from the shear-lag model is faster than that from the Haque’s model. Other than this difference, the stress distribution by the shear-lag model is similar to that by the Haque’s model. It is further noted from our calculations that this discrepancy in the saturated stress length is negligible at higher aspect ratios greater than AR = 300.

Figure 5:

Axial stress distribution in the CNT layer of an SWCNT/epoxy composite obtained from the shear-lag model and Haque’s model, AR = 100.

After the validation of the shear-lag model by FEM and the Haque’s model, load transfer in MWCNT/epoxy composites from the shear-lag model is investigated. Figure 6 shows the axial stress distribution in the epoxy and CNT layers in the five-walled CNT/epoxy composites at AR = 10 and distance s = 0.34 nm. In addition to similar findings in Figure 4, Figure 6 provides general distributions of the induced stress in every layers of the CNT/epoxy composite. It is observed that the stress in epoxy decreases along the REV length direction from applied force positon, σ_k/σ_0m= 1 at y/L = 1 until reaching its own saturation value as σ_s/σ_0m= 0.17 at y/L = 0.5. In contrast, the axial stress in CNT layers increases from σ_k/σ_0m= 0 at y/L = 1 and gradually approaches a saturation value σ_s/σ_0m= 8.8 at y/L = 0.5. Meanwhile, it is worthwhile noting from Figure 4 and Figure 6 that the saturated stress in epoxy and CNT is observed to initiate at the same position, y/L = 0.5 obtained from both shear-lag model and FEM at AR = 10.

Figure 6:

Axial stress distributions of the five-walled CNT/epoxy composite, AR = 10.

After the validation and estimation of Young’s moduli from the present model by that from FEM and experiments followed by an estimation of Young’s moduli is implemented. The effect of CNT geometric structure on the Young’s … ..and oxy is analogous. Then, without loss of generality, the stress distribution in CNT layer only is extracted to examine the influences of CNT and epoxy properties on load transfer in CNT/epoxy composites. Firstly, in order to understand how the aspect ratio affects the load transfer and stress distribution in the composites, a range of aspect ratios from AR = 1 to 1400 is used in simulations from the shear-lag model at the distance between two CNT layers s = 0.34 nm. The effect of the aspect ratio on the saturated stress length as well as the load transfer in the five-walled CNT/epoxy composites from the shear-lag analysis is illustrated in Figure 7. The saturated stress length, L_ss dominates 99 % of composite length at AR = 1000 and above with an interface shear modulus G_iscb = 4.2 GPa in the five-walled CNT composite. It exhibits a well-known finding that the load transfer in CNT/epoxy composites rises along with an increase in the aspect ratio [3–5].

Figure 7:

Effect of the aspect ratio on load transfer in the five-walled CNT/epoxy composite.

The effect of the layer number of CNTs on load transfer in terms of the saturated stress length at AR = 100 and distance s = 0.34 nm is calculated and depicted in Figure 8. In calculations, CNTs associated with single, five and ten layers in composites are examined from the shear-lag model. The results show that the layer number significantly affects the load transfer. The results indicate that the lower load transfer is obtained with a higher layer number. Particularly, an SWCNT shows an excellent reinforcement as the saturated stress length covers 99 % of the half of RVE length, L starting from y/L = 0.7 to 1. The finding implies that the SWCNT/epoxy composites demonstrate very high load transfer from y/L = 0.7 to 1. This phenomenon can be attributed to a perfect bonding between the epoxy and the CNT layer [11]. On the contrary, the saturated stress lengths decease from 94 % to 80 % with an increase in layer numbers from n = 5 to 10, respectively. In addition, based on Figure 8 there is no observation of the axial stress reaching saturation plateau for layer numbers n = 5 and 10. Although the applied load to the MWCNT can be shared by each layer [10], the load transfer from epoxy to CNT layers in MWCNT/epoxy composites is much weaker compared to that of SWCNTs. Therefore, it is preferable to apply CNTs with few layers as reinforcements in composites.

Figure 8:

Effect of CNTs layer number on load transfer in the CNT/epoxy composite, AR = 100.

Figure 10 shows the effect of the CNT volume percentage on load transfer in the five-layered CNT/epoxy composites at AR = 10 based on simulations from the shear-lag model. The initial stress, σ_0m, and all geometry parameters of CNTs are kept unchanged, and only the volume change from epoxy is used for calculations. Based on Figure 10, no obvious influence of CNTs volume on the saturated stress length is observed when the CNT volume fraction increases from 0.5 % to 1 %, corresponding to the epoxy thicknesses ranging from t = 56 nm down to 40 nm, respectively. However, the induced saturated stress in CNT layers decreases from σ_s/σ_0m = 45 to 40.5 with an increment of the CNT volume percentage from 0.5 % to 1 % accordingly. The finding is due to the fact that a higher initial load would be applied to the epoxy when its thickness increases.

Figure 9:

Effect of the epoxy young’s modulus, E_mat, on load transfer in the five-walled CNT/epoxy composite, AR = 9.

In order to study how epoxy affects load transfer and stress distribution in MWCNT composites, two Young’s moduli of epoxy, E_mat= 1 GPa and 10 GPa are used in simulations in the shear-lag model. Figure 1 shows the axial stress distribution in the five-walled CNT/epoxy composites at AR = 10 at the two moduli. The initial stress, σ_σm, is set the same for both cases. From the figure, no noticeable influences of the epoxy property on the saturated stress length are found. This conclusion also supports the findings by the Haque’s model [4]. Nevertheless, normalized saturated stresses of CNT layers decreasing from σ_s /σ_0m = 11.2 to 10 are observed when the epoxy modulus increases from E_mat= 1 GPa to 10 GPa, which is also in agreement with FEM-based findings in literature [6]. It is due to the fact that the stronger epoxy induces a lower epoxy strain which leads to a reduction in axial stress carried by CNT layers.

Figure 10:

Effect of the CNT volume percentage on load transfer in the five-walled CNT/epoxy composite, AR = 10.

The interface shear modulus is used to model the atomistic interaction between epoxy and the CNT layers. The atomistic interactions are different with different epoxy and different bond fillings. Figure 11 is used to understand the effect of the interface shear modulus between the epoxy and CNT layer on load transfer of the five-walled CNT/epoxy composites with an increment of aspect ratio from AR = 1 to 1000 . The figure reveals that no appreciable influence of the interface shear modulus on the saturated stress length and stress distribution trend is found when the interface shear modulus increases from G_ismat= 5 to 30 GPa.

Figure 11:

Effect of the interface shear modulus, G_ismat on load transfer in the five-walled CNT/epoxy composite.

6 Conclusions

A shear-lag model is developed to examine the effect of epoxy and CNT material and geometric properties on load transfer in CNT/epoxy composites under tension. First, a developed FEM is utilized to validate the effectiveness of the shear-lag model. In addition, the Haque’s model is also used to validate the shear-lag model. The three models show good agreements in general and the difference in load transfer among them is negligible at AR = 300 and above. Subsequently, effects of the aspect ratio, volume and Young’s modulus of epoxy and interface bonding stiffness between epoxy and CNT layers are investigated by the shear-lag model. Simulation results indicate that there is no noticeable influence on load transfer from the changes of the epoxy volume and Young’s modulus, E_mat. However, an increase in these two factors may cause higher load bearing on each CNT layer accordingly resulting in the reduction of lifespan and durability of reinforcement. Furthermore, the aspect ratio and layer number of CNTs are found to explicitly affect load transfer in the composites. According to the findings, when aspect ratio rises, load transfer increases accordingly. A highly efficient load transfer for the five-walled CNT/epoxy composites is observed from AR = 1000 or above. Hence, when the half of CNT length is longer than 500 nm, it is suggested to use reinforcements by CNTs with a radius lower than 0.5 nm in nanoscale. The MWCNTs show lower load transfer than its SWCNT counterpart in the composites, and the higher the number of CNT layer, the lower the efficient load transfer. It shows that the SWCNT demonstrates an excellent reinforcement. The investigations on the effect of interface shear modulus, G_ismat on the load transfer in CNT composites is implemented. It is indicated there is no noticeable influence of G_ismat on load transfer in MWCNT/epoxy composites from present predictions.